The park modeled below consists of a rectangular area and 2 semicircular areas. The area of the rectangular part of the park is 60,000 square yards.
Which of the following is closest to the distance along the perimeter of this park?
628 yd
914 yd
1,856 yd
1,228 yd
To find the distance along the perimeter of the park, we need to calculate the sum of the lengths of all sides of the rectangular area and the perimeter of the two semicircular areas.
Given the area of the rectangular part is 60,000 square yards, we can find the dimensions of the rectangular area. Let the length of the rectangle be L and the width be W. Thus, we have:
L * W = 60,000
L = 60,000 / W
The perimeter of the rectangular part is given by:
Perimeter = 2L + 2W
Since the perimeter of a semicircle is half the circumference of the full circle, the perimeter of the two semicircular areas would be equal to the circumference of a full circle with a radius equal to the radius of the semicircle.
Let's assume the width of the rectangular area is 200 yards (which would make the length 300 yards based on the area calculation). This would make the radius of the semicircular areas to be 100 yards.
Now, we can calculate the perimeter of the rectangular area and the perimeter of the two semicircular areas and sum them up to get the total distance along the perimeter of the park:
Perimeter of the rectangle = 2L + 2W
= 2(300) + 2(200)
= 600 + 400
= 1000 yards
Perimeter of each semicircle = πr + 2r
= π(100) + 2(100)
= 100π + 200
Total distance along the perimeter = Perimeter of the rectangle + 2(Perimeter of each semicircle)
= 1000 + 2(100π + 200)
= 1000 + 200π + 400
Approximating the value of π to be 3.14, the total distance along the perimeter would be approximately:
= 1000 + 200(3.14) + 400
≈ 1000 + 628 + 400
≈ 2028 yards
Among the given options, the closest distance to 2028 yards is 1,856 yards. Thus, the closest option to the distance along the perimeter of the park is 1,856 yards.